On a device, e.g. traditional PC, where user input is indirect via a mouse or trackpad in another plane, the cursor is a proxy for the user's focus. Consider that on devices with touchscreens you rarely even see one because the user interacts directly with elements on screen.
The classic arrow cursor indicates to the user where they are on the screen. One might argue that a crosshair cursor would be more visually indicative for this role. However, the arrow is easier to see and is intuitive enough. That ideally should be its only role. But that's not reality! Personally, until seeing this question I never paid attention to what cursors Google was using in Gmail or Microsoft in web Outlook.
Having looked at both web apps and at Whether to use pointer (hand) over a button or just the default cursor it's interesting to see how and where the two cursor styles are applied. They differ in their use but neither seems confusing. (There is one oddity in Gmail: switching between the Primary and Social tabs seems to me a case for the "action" (arrow) cursor but they use the pointer. I suppose it does take you somewhere else).
I didn't need really need to pay attention since there are numerous other visual cues that let me know I can interact with elements on the screen. Buttons look like buttons (and certainly not like the miminimalistminimalist aesthetic that has overtaken iOS where buttonslookbuttons look like links). So perhaps this is something not to worry about too much.Follow convention when it makes sense and otherwise do what you think is right for your unique UX case.
What's important is consistency. Whatever rules you go with, ensure they're applied across the board. If you have something messy or complicated, consider using stylistic differentiation, i.e. if, as in your example, there were two buttons side-by-side with different functionality, don't make them look the same.